Why doesn't the Moon fall to Earth? Research project "Why doesn't the Moon fall to Earth?"

Moon, natural satellite The Earth, in the process of its movement in space, is influenced mainly by two bodies - the Earth and the Sun. At the same time, the sun's gravity is twice as strong as the earth's. Therefore, both bodies (Earth and Moon) revolve around the Sun, being close to each other.

With a twofold predominance of solar gravity over the earth's, the curve of the Moon's motion should be concave in relation to the Sun at all its points. The influence of the nearby Earth, which significantly exceeds the Moon in mass, leads to the fact that the curvature of the lunar heliocentric orbit periodically changes.

The movement of the Earth and the Moon in space and the change in their relative position relative to the Sun are shown in the diagram.

Circulating around the Earth, the Moon moves in orbit at a speed of 1 km/sec, that is, slowly enough not to leave its orbit and “fly” into space, but also fast enough not to fall to the Earth. Directly answering the author of the question, we can say that the Moon will fall to the Earth only if it does not move in orbit, i.e. if external forces (a certain cosmic hand) stop the Moon in its orbital movement, then it naturally will fall to Earth. However, this will release so much energy that talking about the Moon falling on Earth is like solid no need to.

And also by the movement of the Moon.

For clarity, the model of the Moon’s movement in space is simplified. At the same time, we will not lose mathematical and celestial-mechanical rigor if, taking a simpler option as a basis, we do not forget to take into account the influence of numerous factors disturbing the movement.

Assuming the Earth is motionless, we can imagine the Moon as a satellite of our planet, the movement of which obeys Kepler’s laws and occurs along an elliptical orbit. According to a similar scheme, the average value of the eccentricity of the lunar orbit is e = 0.055. The semimajor axis of this ellipse is equal in magnitude to the average distance, i.e. 384,400 km. At the apogee, at the greatest distance, this distance increases to 405,500 km, and at perigee (at the shortest distance) it is 363,300 km. The plane of the lunar orbit is inclined to the ecliptic plane at a certain angle.

Above is a diagram explaining geometric meaning elements of the Moon's orbit.

The elements of the Moon's orbit describe the average, unperturbed motion of the Moon,

However, the influence of the Sun and planets causes the Moon's orbit to change its position in space. The line of nodes moves in the ecliptic plane in the direction reverse movement Moons in orbit. Consequently, the longitude value of the ascending node is constantly changing. The line of nodes completes a full rotation in 18.6 years.

Department of Education of the Kemerovo Municipal District Administration

Xdistrict scientific-practical conference

"World of Discovery"

Section "Geography, geology »

Why doesn't the Moon fall to Earth?

Research project

Semenov Lavr Yurievich,

1st grade student "B"

MBOU "Yagunovskaya Secondary School"

Supervisor:

Kalistratova

Svetlana Borisovna,

teacher primary classes

MBOU "Yagunovskaya Secondary School"

2016

Content

Introduction…………………………………………………………………………………………. 3

Chapter 1. The Moon as a subject of research …………………………………........ 5

1.1. Studying sources……………………………..………………………… 5

1.2. Moon observations...................................................................................... 7

Chapter 2. Organization and results of the study……………………………...9

Conclusion………………………………………………………………………………….. 13

List of references and Internet resources……………………………………….. 14

Introduction

I really like everything related to space. I love watching the stars, finding constellations, so we chose this topic for research.

Kemerovo State University has amazing place- planetarium. It is included in the list of planetariums in Russia, of which there are only 26, as well as in the list of planetariums in the world. "Founder" of our planetarium, teacher, candidate of physical and mathematical sciences of Kemerovo State University, Kuzma Petrovich Matsukov understands “star affairs” better than anyone. The planetarium hosts excursions that reveal the mysteries of space, the birth of the Universe and stars. Here you can see a picture of a real starry sky! Using the starry sky projector under the dome of the planetarium, we can see about five thousand stars, planets, the sun and the moon.

Some planets have many satellites, others have none at all. We decided to figure out what a satellite is. Of course, we were interested in the Moon, since it is a satellite of our Earth.

Having asked Kuzma Petrovich why the Moon always hangs in the sky and does not fly anywhere, they found out that the Earth has amazing property: She attracts everything to herself. But the Moon hangs in the sky and for some reason does not fall to Earth. Why? Let's try to find the answer to this question.

Purpose of the study: reveal why the Moon does not fall to the Earth.

Research objectives:

1. Study various sources on this problem (encyclopedias, Internet), visit the planetarium of Kemerovo State University.

2. Find out how the Moon was formed, how the Moon influences the Earth, what connects the Moon with the Earth.

3. Conduct research and, based on the data obtained, find out why the Moon does not fall on the Earth.

Research hypothesis: It is likely that the Moon will fall if it approaches the Earth. But maybe there is something that keeps the Moon and the Earth at a distance, so the Moon does not fall on the Earth.

Chapter 1. The Moon as a subject of research

1.1 Study of sources

Before we look for the answer to the question “What exactly is the Moon?”, let’s conduct a short survey among adults (5 people) and children (5 people) and find out how deep their knowledge is in this area.

2 people – right;

3 people – wrong.

4 people – right;

1 person – wrong.

Citizens of which country were the first to walk on the moon? (Americans)

0 people – right;

5 people – wrong.

5 people – right;

0 people – wrong.

What was the name of the self-propelled vehicle that traveled on the surface of the Moon? ("Lunokhod")

3 people – right;

2 people – wrong.

5 people – right;

0 people – wrong.

We know that the Earth is a magnet. Why doesn't the Moon, the Earth's satellite, fall to Earth? (It revolves around the Earth)

1 person – right;

4 people – wrong.

4 people – right;

1 person – wrong.

Where did craters come from on the Moon? (From collisions with meteorites)

2 people – right;

3 people – wrong.

5 people - right;

0 people - wrong.

After conducting a survey, we found out that adults can answer questions about the Moon, but children cannot. Therefore, we continued our research.

The word "moon" means "bright". In ancient times, people considered the Moon a goddess - the patroness of the night.

The Moon is the only natural satellite of the Earth. The second brightest object in the earth's sky after the Sun.Currently, astronomers using modern instruments with a laser beam can determine the distance between the Earth and the Moon with an accuracy of several centimeters.The Moon is distant from the Earth at a distance of 384,400 km. Traveling there on foot would take nine years!By car we would need to go to the Moon without stopping for more than six months.

The lunar globe is much smaller than the earth's: in diameter - almost 4 times, and in volume - 49 times. From substance globe 81 balls could be made, each weighing as much as the Moon.

We can only ever see one side of the Moon. A sort of “small” disk, the diameter of which is 3480 km. About half the area of ​​all of Russia.The period of rotation of the Moon around its axis coincides with the period of revolution of the Earth, which is 28 and a half days, so the Moon always faces the Earth with one side.

The Moon rotates around the Earth not strictly in a circle, but in a flattened circle - an ellipse. And when the Moon approaches its maximum, the distance between the Earth and the Moon decreases356,400 kilometers. This minimum approach of the Moon to the Earth is calledperigee . And the maximum distance is calledapogee and equals an integer406,700 kilometers.

There is no atmosphere, so people cannot breathe on the Moon. Surface temperature from −169 °C to +122 °C.

In the old days, gray spots on the Moon were considered seas. It is now known that there is not a drop of water on the Moon, and there is no air shell - atmosphere. The lunar "seas" are deep depressions covered with gray volcanic rocks. Some of the lunar craters were formed when iron or stone bodies - meteorites - fell onto the Moon from interplanetary space. The bright parts of the Moon are its mountainous regions.

We've been to the moon American astronauts. Our lunar rovers controlled from Earth also told us a lot of interesting things about it. Automata and astronauts delivered lunar soil to Earth. The moon is very small, and therefore the force of gravity on it is also small. Astronauts on the Moon weighed about 1/6 of their normal weight on Earth.

The moon is 4.5 billion years old. years - about the same as the Earth. It was formed as a result of a collision of the Earth with one of the small planets. The planet was destroyed, and the Moon formed from its debris and began to gradually move away from the Earth. The distance between it and the Earth is increasing at about the same rate as fingernails grow.

As the Moon orbits the Earth, it exerts gravity on our seas. This attraction causes ebbs and flows.

1.2 Observations of the Moon.

Let's observe the Moon and we will see that its appearance changes every day. At first the crescent is narrow, then the Moon gets fuller and after a few days becomes round. After a few more days, the full Moon gradually becomes smaller and smaller and again becomes like a crescent. The crescent moon is often called the month. If the sickle is turned convex to the left, like the letter “C,” then they say that the Moon is “aging.” 14 days and 19 hours after the full moon old month will disappear completely. The moon is not visible. This phase of the moon is called the “new moon”. Then gradually the Moon, from a narrow sickle turned to the right (if you mentally draw a straight line through the ends of the sickle, you get the letter “P”, i.e. the month is “growing”), turns again into full moon. Sometimes during the new moon the Moon obscures the Sun. At such moments it happens solar eclipse. If the Earth casts a shadow on the Moon during the full moon, then the lunar eclipse. For the Moon to “grow” again, the same period of time is required: 14 days and 19 hours. Changing the appearance of the Moon, i.e. change lunar phases, from full moon to full moon (or from new moon to new moon) occurs every four weeks, more precisely, in 29 and a half days. This is a lunar month. It served as the basis for drawing up the calendar. You can calculate in advance when and how the Moon will be visible, when dark nights, and when they are light. During a full moon, the Moon faces the Earth with its illuminated side, and during a new moon, with its unlit side. The Moon is a solid, cold celestial body that does not emit its own light; it shines in the sky only because it reflects the light of the Sun with its surface. Revolving around the Earth, the Moon turns towards it either with a fully illuminated surface, or with a partially illuminated surface, or with a dark surface. That is why the appearance of the Moon continuously changes throughout the month.

Chapter 2. Organization and results of the study

Today, astronomers imagine the structure solar system like this: in its center is the Sun, and the planets circle around it, as if tied. There are eight of them in total - Mercury, Venus, Earth, Mars, Jupiter, Saturn, Neptune and Uranus. Why, after all, do the planets run around the Sun as if tethered? They are indeed attached, but this connection is invisible. Isaac Newton formulated a very important law - the law universal gravity. He proved that all the bodies of the Universe - the Sun, planets with their satellites, individual stars and star systems- are attracted to each other. The strength of this attraction depends on the size celestial bodies and on the distances between them. The smaller the distance, the stronger the attraction. The greater the distance, the weaker the attraction. Let's conduct a series of experiments.

Experience 1. Let's try to jump in place. What came of it? That's right, we flew up a few centimeters and sank back to the ground. Why don’t we jump and fly high into the sky and then into space? Yes, because we are also tied to our planet by the same force of gravity.

Experience 2. Let's take the ball. It is not flying anywhere, it is at rest, in our hand. We are standing on the floor. We release the ball from our hands and it falls to the floor.

Experience 3. We take a sheet of paper in our hands, throw it up, but it also smoothly falls to the floor.

We observe gravity in nature. We see snow, raindrops falling on the ground. Even icicles grow not upward, but downward, towards the ground.

Conclusion. The earth really holds everything on its surface with a powerful attraction. It holds not only you and me and everything living on Earth, but also all objects, stones, rocks, sands, water of oceans, seas and rivers, the atmosphere surrounding the Earth.

Then why doesn't the Moon fall to Earth?

To begin with, we conducted a survey among children and their parents on the Kemdetki website. The question was asked: “Why do you think the Moon does not fall to Earth?” Here are some of the answers:

1. Dasha, 7 years old: “Because there is air in the sky, and it holds the Moon.”

2. Anya, 7 years old: “Because in zero gravity there is no attraction, it’s a planet!”

3. Olya, 9 years old: “Because the Moon revolves around the Earth in its orbit and cannot leave it.”

4. Matvey, 5 years old: “The Moon is a satellite of the Earth. And in the Earth there is a magnet core and it attracts.”

5. Olya, 5 years old: “Holding on to the air.”

6. Alice, 7 years old: “Because the sky is holding her and she cannot push off...”

7. Roma, 6 years old: “Because she stuck to the night...”

8. Masha, 6 years old: “Where should she fall here? We don’t have enough space here anyway.”

Having studied articles in encyclopedias and the Internet, we found out that the Moon would instantly fall to the Earth if it were stationary. But the Moon does not stand still, it revolves around the Earth. During rotation, a force is formed, which scientists call centripetal, that is, tending towards the center, and centrifugal, running away from the center. We can verify this for ourselves by conducting a series of simple experiments.

Experiment 1. Tie a thread to a regular felt-tip penand let's start rolling it out.The felt-tip pen on the thread will literally pull out of our hand, but the thread will not let go. Centrifugal force acts on the felt-tip pen, trying to throw it away from the center of rotation. So onThe moon is subject to centrifugal force, which prevents it from falling to Earth. Instead, it moves around the Earth on a constant path. If we rotate the felt-tip pen very hard, the thread will break, and if we rotate it slowly, the felt-tip pen will fall. Consequently, if the Moon moved even faster, it would overcome the gravity of the Earth and fly into space; if the Moon moved slower, gravity would pull it towards the Earth.

F1 – centrifugal force (running from the center)

F2- centripetal force (seeking the center)

Experiment 2. Let's take dad's hands, like in a round dance. Without letting go of his hands, we will begin to run around dad, looking into his face, and let dad turn after us. Dad is , and we will be the Moon. If you spin really, really fast, you can even fly without your feet touching the floor. And so that we don’t fly to the wall, dad will have to hold us very tightly. It's the same in heaven. The hands of Father Earth grabbed the Moon tightly and did not let her go.

Experience 3. You can also give an example with the Carousel attraction, which is located in the City Garden of Kemerovo. The rotation speed of the “Carousel” is specially calculated, and if the centrifugal force were less than the tension force of the chain, otherwise it would end in disaster.

Experience 4. Washing machine– the machine gun will also be an example. The laundry that is washed in it is attracted to the walls of its drum when it moves with acceleration, the laundry is spun, and falls only when the drum stops.

Conclusion. That's how the Moon is. If it had not revolved around the Earth, it would probably have fallen onto it. But centrifugal forces prevent her from doing this. And the Moon cannot escape either - the Earth’s gravitational force keeps it in orbit.

Conclusion

So, after studying the literature on this issue and visiting the planetarium of Kemerovo State University, we found out:

That the Moon is the only natural satellite of the Earth.The moon is 4.5 billion years old. years - about the same as the Earth.

Through observations, we noticed that the appearance of the Moon changes every day. Such changes in the shape of the Moon are calledphases.

We also concluded that the Moon is held by the Earth by the force of attraction between the bodies. The force that prevents the Moon from “escaping” during rotation isEarth's gravitational force (centripetal) . And the force that prevents the Moon from falling to Earth isthis is centrifugal force , which occurs when the Moon rotates around the Earth. If the Moon were moving faster, it would overcome the gravity of the Earth and fly into space; if the Moon were moving slower, the force of gravity would pull it towards the Earth.Rotating around the Earth, the Moon moves in orbit at a speed of 1 km/sec, that is, slowly enough not to leave its orbit and “fly” into space, but also fast enough not to fall to the Earth.

Literature and Internet resources

New school encyclopedia“Heavenly Bodies”, M., Rosmen, 2005.

“Why” Children’s Encyclopedia, M., Rosmen, 2005.

“Why doesn’t the Moon fall to Earth?” Zigunenko S.N., Whychkin’s books, 2015.

Rancini. J. “Space. Supernova Atlas of the Universe", M.: Eksmo, 2006.

- “Children!” website for parents of the Kemerovo region.

Wikipedia

Website “For children. Why"

Website “Astronomy and laws of space”

“How simple!”

Ministry of Education of the Russian Federation

Municipal educational institution "Secondary school with. Solodniki."

Abstract

on the topic:

Why doesn't the Moon fall to Earth?

Completed by: 9th grade student,

Feklistov Andrey.

Checked:

Mikhailova E.A.

S. Solodniki 2006

1. Introduction

2. The law of universal gravitation

3. Can the force with which the Earth attracts the Moon be called the weight of the Moon?

4. Is there centrifugal force in the Earth-Moon system, what does it act on?

5. What does the Moon revolve around?

6. Can the Earth and Moon collide? Their orbits around the Sun intersect, and even more than once

7. Conclusion

8. Literature

Introduction

The starry sky has always occupied the imagination of people. Why do stars light up? How many of them shine in the night? Are they far from us? Does the stellar universe have boundaries? Since ancient times, people have thought about these and many other questions, sought to understand and comprehend the structure of big world, in which we live. This opened up a very wide area for exploring the Universe, where gravitational forces play a decisive role.

Among all the forces that exist in nature, the force of gravity differs primarily in that it manifests itself everywhere. All bodies have mass, which is defined as the ratio of the force applied to the body to the acceleration that the body acquires under the influence of this force. The force of attraction acting between any two bodies depends on the masses of both bodies; it is proportional to the product of the masses of the bodies under consideration. In addition, the force of gravity is characterized by the fact that it obeys the law of inverse proportion to the square of the distance. Other forces may depend on distance quite differently; Many such forces are known.

All weighty bodies mutually experience gravity; this force determines the movement of planets around the sun and satellites around the planets. The theory of gravity - a theory created by Newton, stood at the cradle modern science. Another theory of gravity, developed by Einstein, is the greatest achievement of theoretical physics of the 20th century. Over the centuries of human development, people have observed the phenomenon of mutual attraction of bodies and measured its magnitude; they tried to put this phenomenon at their service, to surpass its influence, and, finally, already at the very lately calculate it with extreme accuracy during the first steps deep into the Universe

A widely known story is that Newton's discovery of the law of universal gravitation was prompted by an apple falling from a tree. We don’t know how reliable this story is, but the fact remains that the question: “why doesn’t the moon fall to the earth?” interested Newton and led him to the discovery of the law of universal gravitation. The forces of universal gravity are also called gravitational.

Law of Gravity

Newton's merit lies not only in his brilliant guess about the mutual attraction of bodies, but also in the fact that he was able to find the law of their interaction, that is, a formula for calculating the gravitational force between two bodies.

The law of universal gravitation states: any two bodies attract each other with a force directly proportional to the mass of each of them and inversely proportional to the square of the distance between them

Newton calculated the acceleration imparted to the Moon by the Earth. The acceleration of freely falling bodies at the surface of the earth is equal to 9.8 m/s 2. The Moon is removed from the Earth at a distance equal to approximately 60 Earth radii. Consequently, Newton reasoned, the acceleration at this distance will be: . The Moon, falling with such acceleration, should approach the Earth in the first second by 0.27/2 = 0.13 cm

But the Moon, in addition, moves by inertia in the direction of instantaneous speed, i.e. along a straight line tangent at a given point to its orbit around the Earth (Fig. 1). Moving by inertia, the Moon should move away from the Earth, as calculations show, in one second by 1.3 mm. Of course, we do not observe such a movement in which in the first second the Moon would move radially towards the center of the Earth, and in the second second - along a tangent. Both movements are continuously added. The moon moves along a curved line, close to a circle.

Let us consider an experiment from which we can see how the force of attraction acting on a body at right angles to the direction of motion by inertia transforms rectilinear motion into curvilinear motion (Fig. 2). The ball, having rolled down the inclined chute, continues to move in a straight line by inertia. If you put a magnet on the side, then under the influence of the force of attraction to the magnet, the trajectory of the ball is curved.

No matter how hard you try, you cannot throw a cork ball so that it describes circles in the air, but by tying a thread to it, you can make the ball rotate in a circle around your hand. Experiment (Fig. 3): a weight suspended from a thread passing through a glass tube pulls the thread. The tension force of the thread causes centripetal acceleration, which characterizes the change in linear speed in direction.

The Moon revolves around the Earth, held by gravity. The steel rope that would replace this force would have a diameter of about 600 km. But, despite such a huge gravitational force, the Moon does not fall to the Earth, because it has initial speed and, in addition, moves by inertia.

Knowing the distance from the Earth to the Moon and the number of revolutions of the Moon around the Earth, Newton determined the magnitude of the centripetal acceleration of the Moon.

We got the same number - 0.0027 m/s 2

Stop the gravitational pull of the Moon on the Earth, and it will rush in a straight line into the abyss of outer space. The ball will fly off tangentially (Fig. 3) if the thread holding the ball while rotating in a circle breaks. In the device in Fig. 4, on a centrifugal machine, only a connection (thread) holds the balls in a circular orbit. When the thread breaks, the balls scatter along tangents. It is difficult to catch their rectilinear movement with the eye when they are deprived of connection, but if we make such a drawing (Fig. 5), then it follows from it that the balls will move rectilinearly, tangentially to the circle.

Stop the movement by inertia - and the Moon would fall to the Earth. The fall would have lasted four days, nineteen hours, fifty-four minutes, fifty-seven seconds, as Newton calculated.

Using the formula of the law of universal gravitation, you can determine with what force the Earth attracts the Moon: where G-gravitational constant, T 1 and m 2 are the masses of the Earth and the Moon, r is the distance between them. Substituting specific data into the formula, we obtain the value of the force with which the Earth attracts the Moon and it is approximately 2 10 17 N

The law of universal gravitation applies to all bodies, which means that the Sun also attracts the Moon. Let's count with what force?

The mass of the Sun is 300,000 times the mass of the Earth, but the distance between the Sun and the Moon is 400 times greater than the distance between the Earth and the Moon. Therefore, in the formula the numerator will increase by 300,000 times, and the denominator will increase by 400 2, or 160,000 times. The gravitational force will be almost twice as strong.

But why doesn't the Moon fall on the Sun?

The Moon falls on the Sun in the same way as on the Earth, that is, only enough to remain at approximately the same distance while revolving around the Sun.

The Earth and its satellite, the Moon, revolve around the Sun, which means the Moon also revolves around the Sun.

The following question arises: the Moon does not fall to the Earth, because, having an initial speed, it moves by inertia. But according to Newton's third law, the forces with which two bodies act on each other are equal in magnitude and opposite in direction. Therefore, with the same force with which the Earth attracts the Moon, with the same force the Moon attracts the Earth. Why doesn't the Earth fall on the Moon? Or does it also revolve around the Moon?

The fact is that both the Moon and the Earth revolve around a common center of mass, or, to simplify, one might say, around a common center of gravity. Remember the experiment with balls and a centrifugal machine. The mass of one of the balls is twice the mass of the other. In order for the balls connected by a thread to remain in equilibrium about the axis of rotation during rotation, their distances from the axis, or center of rotation, must be inversely proportional to the masses. The point or center around which these balls revolve is called the center of mass of the two balls.

Newton's third law is not violated in the experiment with balls: the forces with which the balls pull each other towards a common center of mass are equal. In the Earth-Moon system, the common center of mass revolves around the Sun.

Is it possible the force with which the Earth attracts Lu Well, call it the weight of the Moon?

No, you can't. We call the weight of a body the force caused by the gravity of the Earth with which the body presses on some support: a scale, for example, or stretches the spring of a dynamometer. If you place a stand under the Moon (on the side facing the Earth), the Moon will not put pressure on it. Luna would not stretch the dynamometer spring even if they could suspend it. The entire effect of the force of attraction of the Moon by the Earth is expressed only in keeping the Moon in orbit, in imparting centripetal acceleration to it. We can say about the Moon that in relation to the Earth it is weightless in the same way that objects in a spaceship-satellite are weightless when the engine stops working and only the force of gravity towards the Earth acts on the ship, but this force cannot be called weight. All objects released from the hands of the astronauts (pen, notepad) do not fall, but float freely inside the cabin. All bodies located on the Moon, in relation to the Moon, are, of course, weighty and will fall to its surface if they are not held by something, but in relation to the Earth these bodies will be weightless and cannot fall to the Earth.

Is there centrifugal force in system Earth - Moon, what does it act on?

In the Earth-Moon system, the forces of mutual attraction between the Earth and the Moon are equal and oppositely directed, namely towards the center of mass. Both of these forces are centripetal. There is no centrifugal force here.

The distance from the Earth to the Moon is approximately 384,000 km. The ratio of the mass of the Moon to the mass of the Earth is 1/81. Consequently, the distances from the center of mass to the centers of the Moon and Earth will be inversely proportional to these numbers. Dividing 384,000 km at 81, we get approximately 4,700 km. This means that the center of mass is at a distance of 4,700 km from the center of the Earth.

The radius of the Earth is about 6400 km. Consequently, the center of mass of the Earth-Moon system lies inside the globe. Therefore, if we do not strive for accuracy, we can talk about the Moon’s revolution around the Earth.

It is easier to fly from Earth to the Moon or from the Moon to Earth, because... It is known that in order for a rocket to become an artificial satellite of the Earth, it must be given an initial speed of ≈ 8 km/sec. In order for the rocket to leave the Earth's sphere of gravity, the so-called second escape velocity is needed, equal to 11.2 km/sec. To launch rockets from the Moon, you need a lower speed because... The gravity on the Moon is six times less than on Earth.

The bodies inside the rocket become weightless from the moment the engines stop working and the rocket flies freely in orbit around the Earth, while being in the Earth's gravitational field. During free flight around the Earth, both the satellite and all objects in it relative to the Earth's center of mass move with the same centripetal acceleration and therefore weightless.

How did the balls not connected by a thread move on a centrifugal machine: along a radius or along a tangent to a circle? The answer depends on the choice of the reference system, i.e., relative to which reference body we will consider the movement of the balls. If we take the table surface as the reference system, then the balls moved along tangents to the circles they described. If we take the rotating device itself as the reference system, then the balls moved along a radius. Without indicating a reference system, the question of motion makes no sense at all. To move means to move relative to other bodies, and we must necessarily indicate which ones.

What does the Moon revolve around?

If we consider the movement relative to the Earth, then the Moon revolves around the Earth. If we take the Sun as the body of reference, then - around the Sun.

Could the Earth and Moon collide? Their shout bits around the Sun intersect, and more than once .

Of course not. A collision would only be possible if the Moon's orbit relative to the Earth intersected the Earth. When the position of the Earth or the Moon is at the intersection of the shown orbits (relative to the Sun), the distance between the Earth and the Moon is on average 380,000 km. To understand this better, let's draw the following. The Earth's orbit is depicted as an arc of a circle with a radius of 15 cm (the distance from the Earth to the Sun is known to be 150,000,000 km). On an arc equal to part of the circle (the monthly path of the Earth), I marked five points at equal distances, counting the outermost ones. These points will be the centers of the lunar orbits relative to the Earth in successive quarters of the month. The radius of the lunar orbits cannot be depicted on the same scale as the Earth's orbit, since it will be too small. To draw the lunar orbits, you need to increase the selected scale by about ten times, then the radius of the lunar orbit will be about 4 mm. After that indicated the position of the Moon in each orbit, starting with the full moon, and connected the marked points with a smooth dotted line.

The main task was to separate the reference bodies. In an experiment with a centrifugal machine, both bodies of reference are simultaneously projected onto the plane of the table, so it is very difficult to focus attention on one of them. This is how we solved our problem. A ruler made of thick paper (it can be replaced with a strip of tin, plexiglass, etc.) will serve as a rod along which a cardboard circle resembling a ball slides. The circle is double, glued along the circumference, but on two diametrically opposite sides there are slits through which a ruler is threaded. Holes are made along the axis of the ruler. The reference bodies are a ruler and a sheet of blank paper, which we attached to a sheet of plywood with buttons so as not to spoil the table. Having placed the ruler on a pin, like on an axle, we stuck the pin into the plywood (Fig. 6). When the ruler was rotated at equal angles, successive holes appeared on the same straight line. But when the ruler was turned, a cardboard circle slid along it, the successive positions of which had to be marked on paper. For this purpose, a hole was also made in the center of the circle.

With each rotation of the ruler, the position of the center of the circle was marked on paper with the tip of a pencil. When the ruler had passed through all the positions previously planned for it, the ruler was removed. By connecting the marks on the paper, we made sure that the center of the circle moved relative to the second reference body in a straight line, or rather, tangent to the initial circle.

But while working on the device, I made several interesting discoveries. Firstly, with uniform rotation of the rod (ruler), the ball (circle) moves along it not uniformly, but accelerated. By inertia, a body must move uniformly and in a straight line - this is a law of nature. But did our ball move only by inertia, i.e. freely? No! The rod pushed him and gave him acceleration. This will be clear to everyone if you refer to the drawing (Fig. 7). On a horizontal line (tangent) with points 0, 1, 2, 3, 4 The positions of the ball are marked if it were to move completely freely. The corresponding positions of the radii with the same digital designations show that the ball is moving at an accelerated rate. The acceleration of the ball is imparted by the elastic force of the rod. In addition, friction between the ball and the rod provides resistance to movement. If we assume that the friction force is equal to the force that imparts acceleration to the ball, the movement of the ball along the rod should be uniform. As can be seen from Figure 8, the movement of the ball relative to the paper on the table is curvilinear. In drawing lessons we were told that such a curve is called the “Archimedes spiral”. The profile of cams in some mechanisms is drawn along such a curve when they want to transform a uniform rotational movement into a uniform translational movement. If you put two such curves next to each other, the cam will get a heart-shaped shape. With uniform rotation of a part of this shape, the rod resting on it will perform a forward-return motion. I made a model of such a cam (Fig. 9) and a model of the mechanism for uniformly winding threads onto a spool (Fig. 10).

I did not make any discoveries while completing the task. But I learned a lot while making this chart (Figure 11). It was necessary to correctly determine the position of the Moon in its phases, to think about the direction of movement of the Moon and the Earth in their orbits. There are inaccuracies in the drawing. I'll tell you about them now. The selected scale incorrectly depicts the curvature of the lunar orbit. It must always be concave in relation to the Sun, that is, the center of curvature must be inside the orbit. In addition, there are not 12 lunar months in a year, but more. But one-twelfth of a circle is easy to construct, so I conventionally assumed that there are 12 lunar months in a year. And finally, it is not the Earth itself that revolves around the Sun, but the common center of mass of the Earth-Moon system.

Conclusion

One of bright examples achievements of science, one of the evidence of the unlimited cognition of nature was the discovery of the planet Neptune through calculations - “at the tip of a pen.”

Uranus, the planet next to Saturn, which for many centuries was considered the most distant of the planets, was discovered by W. Herschel at the end of the 18th century. Uranus is hardly visible to the naked eye. By the 40s of the XIX century. accurate observations showed that Uranus deviates scarcely perceptibly from the path it should follow, taking into account the disturbances from all the known planets. Thus the theory of the motion of celestial bodies, so strict and precise, was put to the test.

Le Verrier (in France) and Adams (in England) suggested that if disturbances from outside famous planets do not explain the deviation in the movement of Uranus, which means that the attraction of an as yet unknown body acts on it. They almost simultaneously calculated where behind Uranus there should be an unknown body producing these deviations with its attraction. They calculated the orbit of the unknown planet, its mass and indicated the place in the sky where given time there must have been an unknown planet. This planet was found through a telescope at the place they indicated in 1846. It was named Neptune. Neptune is not visible to the naked eye. Thus, the disagreement between theory and practice, which seemed to undermine the authority of materialist science, led to its triumph.

References:

1. M.I. Bludov - Conversations on Physics, part one, second edition, revised, Moscow “Enlightenment” 1972.

2. B.A. Vorontsov-Velyamov – Astronomy! 1st grade, 19th edition, Moscow “Enlightenment” 1991.

3. A.A. Leonovich - I explore the world, Physics, Moscow AST 1998.

4. A.V. Peryshkin, E.M. Gutnik - Physics 9th grade, Publishing house "Drofa" 1999.

5. Ya.I. Perelman – Entertaining physics, book 2, 19th edition, Nauka publishing house, Moscow 1976.

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Student . A widely known story is that Newton's discovery of the law of universal gravitation was prompted by an apple falling from a tree. We don’t know how reliable this story is, but the fact remains that the question that we are here to discuss today: “Why doesn’t the Moon fall to the Earth?” interested Newton and led him to the discovery of the law of gravity. Newton argued that between the Earth and everyone material bodies There is a gravitational force that is inversely proportional to the square of the distance.

Newton calculated the acceleration imparted to the Moon by the Earth. The acceleration of freely falling bodies near the Earth's surface is g=9.8 m/s 2 . The Moon is removed from the Earth at a distance equal to approximately 60 Earth radii. Therefore, Newton reasoned, the acceleration at this distance will be: . The Moon, falling with such acceleration, should approach the Earth by 0.0013 m in the first second. But the Moon, in addition, moves by inertia in the direction of instantaneous speed, i.e. along a straight line tangent at a given point to its orbit around the Earth (Fig. 25).

Moving by inertia, the Moon should move away from the Earth, as calculations show, in one second by 1.3 mm. Of course, such a movement in which in the first second the Moon would move radially towards the center of the Earth, and in the second second – along a tangent, does not actually exist. Both movements are continuously added. As a result, the Moon moves along a curved line, close to a circle.

Let us carry out an experiment from which we can see how the force of attraction acting on a body at right angles to the direction of its movement transforms rectilinear movement into a curvilinear one. The ball, having rolled down the inclined chute, continues to move in a straight line by inertia. If you put a magnet on the side, then under the influence of the force of attraction to the magnet, the trajectory of the ball is curved (Fig. 26).

The Moon revolves around the Earth, held by gravity. A steel cable that could hold the Moon in orbit would have to have a diameter of about 600 km. But, despite such a huge gravitational force, the Moon does not fall to the Earth, because, having an initial speed, it moves by inertia.

Knowing the distance from the Earth to the Moon and the number of revolutions of the Moon around the Earth, Newton determined the centripetal acceleration of the Moon. The result is a number we already know: 0.0027 m/s2.
Stop the force of attraction of the Moon to the Earth - and the Moon will rush in a straight line into the abyss outer space. Thus, in the device shown in Figure 27, the ball will fly off tangentially if the thread holding the ball on the circle breaks. In the device you know on a centrifugal machine (Fig. 28), only a connection (thread) holds the balls in a circular orbit.

When the thread breaks, the balls scatter along tangents. It is difficult to catch their rectilinear movement with the eye when they are deprived of connection, but if we make a drawing (Fig. 29), we will see that the balls move rectilinearly, tangentially to the circle.

If the inertial movement had stopped, the Moon would have fallen to the Earth. The fall would have lasted four days, nineteen hours, fifty-four minutes, fifty-seven seconds, as Newton calculated.

A teacher present at a circle lesson. The report is finished. Who has questions?

Question . With what force does the Earth attract the Moon?

Student . This can be determined by the formula expressing the law of gravity: , where G is the gravitational constant, M and m are the masses of the Earth and the Moon, r is the distance between them. I was expecting this question and did the calculation in advance. The Earth attracts the Moon with a force of about 2 * 10 20 N.

Question . The law of universal gravitation applies to all bodies, which means that the Sun also attracts the Moon. I wonder with what force?

Answer . The mass of the Sun is 300,000 times the mass of the Earth, but the distance between the Sun and the Moon is 400 times greater than the distance between the Earth and the Moon. Therefore, in the formula the numerator will increase by 300,000 times, and the denominator will increase by 400 2, or 160,000 times. The gravitational force will be almost twice as strong.

Question . Why doesn't the Moon fall on the Sun?

Answer . The Moon falls on the Sun in the same way as on the Earth, that is, only enough to remain at approximately the same distance while revolving around the Sun.

- Around the Earth!

– Incorrect, not around the Earth, but around the Sun. The Earth and its satellite, the Moon, revolve around the Sun, which means that the Moon also revolves around the Sun.

Question . The Moon does not fall to the Earth because, having an initial speed, it moves by inertia. But according to Newton's third law, the forces with which two bodies act on each other are equal in magnitude and opposite in direction. Therefore, with the same force with which the Earth attracts the Moon, with the same force the Moon attracts the Earth. Why doesn't the Earth fall on the Moon? Or does it orbit the Moon?

Teacher . The fact is that both the Moon and the Earth revolve around a common center of mass. Remember the experiment with balls and a centrifugal machine. The mass of one of the balls is twice the mass of the other. In order for the balls connected by a thread to remain in equilibrium about the axis of rotation during rotation, their distances from the axis, or center of rotation, must be inversely proportional to the masses. The point around which these balls revolve is called the center of mass of the two balls.

Newton's third law is not violated in the experiment with balls: the forces with which the balls pull each other towards a common center of mass are equal. The common center of mass of the Earth and Moon revolves around the Sun.

Question . Can the force with which the Earth attracts the Moon be called the weight of the Moon?

Student . No, you can't! We call the weight of a body the force caused by the gravity of the Earth with which the body presses on some support, for example, a scale, or stretches the spring of a dynamometer. If you place a stand under the Moon (on the side facing the Earth), the Moon will not put pressure on it. The Moon would not stretch the spring of the dynamometer if we could suspend it. The entire effect of the force of attraction of the Moon by the Earth is expressed only in keeping the Moon in orbit, in imparting centripetal acceleration to it. We can say about the Moon that in relation to the Earth it is weightless in the same way that objects in a spaceship-satellite are weightless when the engine stops working and only the force of gravity towards the Earth acts on the ship.

Question . Where is the center of mass of the Earth-Moon system?

Answer . The distance from the Earth to the Moon is 384,000 km. The ratio of the mass of the Moon to the mass of the Earth is 1:81. The distances from the center of mass to the centers of the Moon and Earth will be inversely proportional to these numbers. Dividing 384,000 km by 82, we get approximately 4,700 km. This means that the center of mass is located at a distance of 4700 km from the center of the Earth.

– What is the radius of the Earth?

– About 6400 km.

– Consequently, the center of mass of the Earth-Moon system lies inside the globe (Fig. 30, point O). Therefore, if we do not strive for accuracy, we can talk about the Moon’s revolution around the Earth.

Question . What is easier: to fly from Earth to the Moon or from the Moon to Earth?

Answer . In order for a rocket to become an artificial Earth satellite, it must be given an initial speed of approximately 8 km/s. In order for the rocket to leave the Earth's sphere of gravity, the so-called second escape velocity is needed, equal to 11.2 km/s. To launch rockets from the Moon, you need a lower speed: after all, the force of gravity on the Moon is six times less than on Earth.

Question . I don't understand very well why the bodies inside the rocket have no weight. Maybe it is only at that point on the way to the Moon at which the force of gravity on the Moon is balanced by the force of gravity on the Earth?

Teacher . No. The bodies inside the rocket become weightless from the moment the engines stop working and the rocket begins a free flight in orbit around the Earth, while being in the Earth's gravitational field. During free flight around the Earth, both the satellite and all objects in it relative to the Earth's center of mass move with the same centripetal acceleration and are therefore weightless.

1st question. How did the balls not connected by a thread move on a centrifugal machine: along a radius or along a tangent to a circle?

The answer depends on the choice of the reference system, i.e. on the choice of the body relative to which we are considering the motion of the balls. If we take the table surface as the reference system, then the balls moved along tangents to the circles they described. If we take the rotating device itself as the reference system, then the balls moved along a radius. Without indicating the reference system, the question about the nature of the movement does not make sense. To move means to move relative to other bodies, and we must necessarily indicate which ones.

2nd question. What does the Moon revolve around?

If we consider the movement relative to the Earth, the Moon revolves around the Earth. If we take the Sun as the body of reference, then it will be around the Sun. Let me explain what I said with a drawing from the book “ Entertaining astronomy» Perelman (Fig. 31). Tell me, relative to which body the movement of celestial bodies is shown here.

– Regarding the Sun.

- Right. But it is easy to notice that the Moon is constantly changing its position relative to the Earth.

Teacher . Of course they can't. At the position of the Earth or the Moon (note that I say “or”, not “and”) at the point of intersection of the orbits shown, the distance between the Earth and the Moon is 380,000 km. To better understand this, draw a diagram of this complex movement for your next lesson. Draw the Earth's orbit as an arc of a circle with a radius of 15 cm (the distance from the Earth to the Sun, as is known, is 150,000,000 km). On an arc equal to 1/12 of a circle (monthly path of the Earth), mark on equal distances five points, including the outer ones. These points will be the centers of the lunar orbits relative to the Earth in successive quarters of the month. The radius of the lunar orbits cannot be depicted on the same scale as the Earth's orbit, since it will be too small. To draw the lunar orbits, you need to increase the selected scale by about ten times, then the radius of the lunar orbit will be about 4 mm. Indicate the position of the Moon in each orbit, starting with the full moon, and connect the marked points with a smooth dotted line.

At the next class, one of the students showed the required diagram (Fig. 32).

Story from a student drawing a diagram: “I learned a lot while drawing this diagram. It was necessary to correctly determine the position of the Moon in its phases, to think about the direction of movement of the Moon and the Earth in their orbits. There are inaccuracies in the drawing. I'll tell you about them now. The selected scale incorrectly depicts the curvature of the lunar orbit. It must always be concave in relation to the Sun, that is, the center of curvature must be inside the orbit. In addition, there are not 12 lunar months in a year, but more. But one twelfth of a circle is easy to construct, so I conventionally assumed that there are 12 lunar months in a year. And finally, it is not the Earth itself that revolves around the Sun, but the common center of mass of the Earth-Moon system.”

One ancient Greek, supposedly Plutarch, said: as soon as the Moon slows down, it will immediately fall to the Earth, like a stone released from a sling. This was said back when stars, not meteorites, were falling. Seventeen centuries later, Galileo, armed not only with the art of reasonable generalizations, but also with a telescope, continued: The Moon, they say, does not slow down because it moves by inertia, and obviously nothing prevents this movement. He said how he cut it. Another two hundred years later, Newton added his two cents: they say, dear ones, if the Moon moved only by inertia, it would move in a straight line, having long ago disappeared into the abyss of the Universe; The Earth and the Moon are held next to each other by the force of mutual gravity, forcing the latter to move in a circle. Moreover, he said, gravity, being most likely the root cause of any movement in the Universe, is capable of even accelerating the slightly slower run of the Moon in certain sections of the elliptical (Kepler) orbit... A hundred years later, Cavendish, using lead balls and torsion balances, proved the existence of mutual force gravity. That's it. Therefore, it is inertia and gravity, forcing the Moon to move in a closed orbit, that are the reasons that prevent the Moon from falling to the Earth. In short, if the gravitational mass of the Earth suddenly increases, then the Moon will only move away from it in its higher orbit. But... The satellites of the planets cannot have any closed orbits - circular or elliptical. Now we will look at the joint “fall” of the Earth and the Moon on the Sun and make sure of this. So, the Earth and the Moon have been “falling” together in the gravitational space of the Sun for about 4 billion years. At the same time, the speed of the Earth relative to the Sun is approximately 30 km/s, and the Moon – 31. In 30 days, the Earth travels along its trajectory 77.8 million km (30 x 3600 x 24 x 30), and the Moon – 80.3. 80.3 – 77.8 = 2.5 million km. The radius of the Moon's orbit is approximately 400,000 km. Therefore, the circumference of the Moon’s orbit is 400,000 x 2 x 3.14 = 2.5 million km. Only in our reasoning, 2.5 million km is already the “curvature” of the almost straight trajectory of the Moon. A large-scale display of the trajectories of the Earth and the Moon may also look like this: if there are 1 million km in one cell, then the path traveled by the Earth and the Moon in a month will not fit into the entire spread of a notebook in a cell, while the maximum distance between the Moon’s trajectory and the Earth’s trajectory in the full moon and new moon phases it will be equal to only 2 millimeters. However, you can take a segment of arbitrary length, indicating the path of the Earth, and draw the movement of the Moon over a month. The movement of the Earth and the Moon occurs from right to left, that is, counterclockwise. If we have the Sun somewhere at the bottom of the picture, then on the right side of the picture we will mark the Moon in the full moon phase with a dot. Let the Earth at this time be exactly under this point. In 15 days, the Moon will be in the new moon phase, that is, right in the middle of our segment and just under the Earth in the figure. On the left side of the figure we again denote the positions of the Moon and Earth in the full moon phase with dots. Over the course of a month, the Moon crosses the trajectory of the Earth twice at the so-called nodes. The first node will be approximately 7.5 days from the full moon phase. From the Earth at this time, just half of the lunar disk is visible. This phase is called the first quarter, since by this time the Moon has completed a quarter of its monthly path. The second time the Moon crosses the Earth's trajectory is in the last quarter, that is, approximately 7.5 days from the new moon phase. Did you draw it? Here’s what’s interesting: the Moon at the first quarter node is 400,000 km ahead of the Earth, and at the last quarter node it is already 400,000 km behind it. It turns out that the Moon “along the upper crest of the wave” moves with acceleration, and “along the lower crest” - with deceleration; the path of the Moon from the last quarter node to the first quarter node is 800,000 km longer. Of course, the Moon in its movement along the “upper arc” does not accelerate spontaneously, it is the Earth with its gravitational mass that captures it and, as it were, throws it over itself. It is this property of moving planets - to capture and throw - that is used to accelerate space probes during the so-called gravitational maneuver. If the probe crosses the path of the planet in front of it, then we have a gravitational maneuver with the probe slowing down. It's simple. The full moon phase repeats after 29 days, 12 hours and 44 minutes. This is the synodic period of the Moon's revolution. Theoretically, the Moon should complete its orbital journey in 27 days, 7 hours and 43 minutes. This is the sidereal period of revolution. The “inconsistency” of two days in textbooks is explained by the movement of the Earth and the Moon per month relative to the round Sun. We explained this by the absence of any orbit on the Moon. So, Newton explained the “non-fall” of the Moon on the Earth by its temporary accelerations when moving along an elliptical orbit. We, I think, explained this even more simply. And most importantly - more correctly. Viktor Babintsev